Notes on Noncommutative Instantons

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We study in detail the ADHM construction of U(N) instantons on noncommutative Euclidean space-time RNC⁴ and noncommutative space RNC² x R². We point out that the completeness condition in the ADHM construction could be invalidated in certain circumstances. When this happens, regular instanton configuration may not exist even if the ADHM constraints are satisfied. Some of the existing solutions in the literature indeed violate the completeness condition and hence are not correct. We present alternative solutions for these cases. In particular, we show for the first time how to construct explicitly regular U(N) instanton solutions on RNC⁴ and on RNC² x R². We also give a simple general argument based on the Corrigan's identity that the topological charge of noncommutative regular instantons is always an integer.